Under what conditions is the homology of a dg coalgebra a graded coalgebra?
I'm trying to get a feel for some differential graded (dg) structures.
Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $Delta : C to C otimes C$ and a counit $varepsilon : C to k$ satisfying the usual axioms.
I'm interested in some sufficient conditions for the coalgebra structure on $C$ to induce coalgebra structure on the homology $H(C)$ (which is a graded $k$-module).
I guess if $k$ is a field (or a ring for which the relevant $operatorname{Tor}$'s in the Künneth sequence vanish) then the map $H(C)otimes H(C) to H(C otimes C)$ is an isomorphism, so the inverse can be used to define a coalgebra structure on $H(C)$. What if $k$ is a more complicated ring?
What about conditions "about $C$" instead of conditions "about $k$"?
Is it correct that when dealing with a product-type structure (e.g. a dg algebra or a dg Lie algebra) then no use of the Künneth formula is needed to induce the product-type structure on homology?
Many thanks!
abstract-algebra algebraic-topology homological-algebra coalgebras
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
asked Jul 14 '15 at 11:21
user50948
7941415
add a comment |
I'm trying to get a feel for some differential graded (dg) structures.
Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $Delta : C to C otimes C$ and a counit $varepsilon : C to k$ satisfying the usual axioms.
I'm interested in some sufficient conditions for the coalgebra structure on $C$ to induce coalgebra structure on the homology $H(C)$ (which is a graded $k$-module).
I guess if $k$ is a field (or a ring for which the relevant $operatorname{Tor}$'s in the Künneth sequence vanish) then the map $H(C)otimes H(C) to H(C otimes C)$ is an isomorphism, so the inverse can be used to define a coalgebra structure on $H(C)$. What if $k$ is a more complicated ring?
What about conditions "about $C$" instead of conditions "about $k$"?
Is it correct that when dealing with a product-type structure (e.g. a dg algebra or a dg Lie algebra) then no use of the Künneth formula is needed to induce the product-type structure on homology?
Many thanks!
abstract-algebra algebraic-topology homological-algebra coalgebras
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
asked Jul 14 '15 at 11:21
user50948
7941415
add a comment |
I'm trying to get a feel for some differential graded (dg) structures.
Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $Delta : C to C otimes C$ and a counit $varepsilon : C to k$ satisfying the usual axioms.
I'm interested in some sufficient conditions for the coalgebra structure on $C$ to induce coalgebra structure on the homology $H(C)$ (which is a graded $k$-module).
I guess if $k$ is a field (or a ring for which the relevant $operatorname{Tor}$'s in the Künneth sequence vanish) then the map $H(C)otimes H(C) to H(C otimes C)$ is an isomorphism, so the inverse can be used to define a coalgebra structure on $H(C)$. What if $k$ is a more complicated ring?
What about conditions "about $C$" instead of conditions "about $k$"?
Is it correct that when dealing with a product-type structure (e.g. a dg algebra or a dg Lie algebra) then no use of the Künneth formula is needed to induce the product-type structure on homology?
Many thanks!
abstract-algebra algebraic-topology homological-algebra coalgebras
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
asked Jul 14 '15 at 11:21
user50948
7941415
I'm trying to get a feel for some differential graded (dg) structures.
Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $Delta : C to C otimes C$ and a counit $varepsilon : C to k$ satisfying the usual axioms.
I'm interested in some sufficient conditions for the coalgebra structure on $C$ to induce coalgebra structure on the homology $H(C)$ (which is a graded $k$-module).
I guess if $k$ is a field (or a ring for which the relevant $operatorname{Tor}$'s in the Künneth sequence vanish) then the map $H(C)otimes H(C) to H(C otimes C)$ is an isomorphism, so the inverse can be used to define a coalgebra structure on $H(C)$. What if $k$ is a more complicated ring?
What about conditions "about $C$" instead of conditions "about $k$"?
Is it correct that when dealing with a product-type structure (e.g. a dg algebra or a dg Lie algebra) then no use of the Künneth formula is needed to induce the product-type structure on homology?
Many thanks!
abstract-algebra algebraic-topology homological-algebra coalgebras
abstract-algebra algebraic-topology homological-algebra coalgebras
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
asked Jul 14 '15 at 11:21
user50948
7941415
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
asked Jul 14 '15 at 11:21
user50948
7941415
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
edited Nov 29 at 22:11
Arnaud Mortier
19.8k22260
19.8k22260
asked Jul 14 '15 at 11:21
user50948
7941415
asked Jul 14 '15 at 11:21
user50948
7941415
asked Jul 14 '15 at 11:21
user50948
7941415
7941415
add a comment |
add a comment |
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